We consider the problem of cost sensitive multiclass classification, where we would like to increase the sensitivity of an important class at the expense of a less important one. We adopt an {\em apportioned margin} framework to address this problem, which enables an efficient margin shift between classes that share the same boundary. The decision boundary between all pairs of classes divides the margin between them in accordance to a given prioritization vector, which yields a tighter error bound for the important classes while also reducing the overall out-of-sample error. In addition to demonstrating an efficient implementation of our framework, we derive generalization bounds, demonstrate Fisher consistency, adapt the framework to Mercer's kernel and to neural networks, and report promising empirical results on all accounts.
Adversarial robustness has become an important research topic given empirical demonstrations on the lack of robustness of deep neural networks. Unfortunately, recent theoretical results suggest that adversarial training induces a strict tradeoff between classification accuracy and adversarial robustness. In this paper, we propose and then study a new regularization for any margin classifier or deep neural network. We motivate this regularization by a novel generalization bound that shows a tradeoff in classifier accuracy between maximizing its margin and average margin. We thus call our approa…
We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic…
Boosting is one of the most successful ideas in machine learning. The most well-accepted explanations for the low generalization error of boosting algorithms such as AdaBoost stem from margin theory. The study of margins in the context of boosting algorithms was initiated by Schapire, Freund, Bartlett and Lee (1998) and has inspired numerous boosting algorithms and generalization bounds. To date, the strongest known generalization (upper bound) is the $k$th margin bound of Gao and Zhou (2013). Despite the numerous generalization upper bounds that have been proved over the last two decades, not…
University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）；ServiceNow Research（ServiceNow研究院）
We study the generalisation properties of majority voting on finite ensembles of classifiers, proving margin-based generalisation bounds via the PAC-Bayes theory. These provide state-of-the-art guarantees on a number of classification tasks. Our central results leverage the Dirichlet posteriors studied recently by Zantedeschi et al. [2021] for training voting classifiers; in contrast to that work our bounds apply to non-randomised votes via the use of margins. Our contributions add perspective to the debate on the "margins theory" proposed by Schapire et al. [1998] for the generalisation of en…
Logistic models are commonly used for binary classification tasks. The success of such models has often been attributed to their connection to maximum-likelihood estimators. It has been shown that gradient descent algorithm, when applied on the logistic loss, converges to the max-margin classifier (a.k.a. hard-margin SVM). The performance of the max-margin classifier has been recently analyzed. Inspired by these results, in this paper, we present and study a more general setting, where the underlying parameters of the logistic model possess certain structures (sparse, block-sparse, low-rank, e…
Radboud University（拉德堡德大学）；Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）
Consider a classification problem where we have both labeled and unlabeled data available. We show that for linear classifiers defined by convex margin-based surrogate losses that are decreasing, it is impossible to construct any semi-supervised approach that is able to guarantee an improvement over the supervised classifier measured by this surrogate loss on the labeled and unlabeled data. For convex margin-based loss functions that also increase, we demonstrate safe improvements are possible.
Stability is an important aspect of a classification procedure because unstable predictions can potentially reduce users' trust in a classification system and also harm the reproducibility of scientific conclusions. The major goal of our work is to introduce a novel concept of classification instability, i.e., decision boundary instability (DBI), and incorporate it with the generalization error (GE) as a standard for selecting the most accurate and stable classifier. Specifically, we implement a two-stage algorithm: (i) initially select a subset of classifiers whose estimated GEs are not signi…
Boosting is one of the most successful ideas in machine learning, achieving great practical performance with little fine-tuning. The success of boosted classifiers is most often attributed to improvements in margins. The focus on margin explanations was pioneered in the seminal work by Schapire et al. (1998) and has culminated in the $k$'th margin generalization bound by Gao and Zhou (2013), which was recently proved to be near-tight for some data distributions (Gronlund et al. 2019). In this work, we first demonstrate that the $k$'th margin bound is inadequate in explaining the performance of…
We address the problem of aggregating an ensemble of predictors with known loss bounds in a semi-supervised binary classification setting, to minimize prediction loss incurred on the unlabeled data. We find the minimax optimal predictions for a very general class of loss functions including all convex and many non-convex losses, extending a recent analysis of the problem for misclassification error. The result is a family of semi-supervised ensemble aggregation algorithms which are as efficient as linear learning by convex optimization, but are minimax optimal without any relaxations. Their de…
A problem of bounding the generalization error of a classifier f in H, where H is a "base" class of functions (classifiers), is considered. This problem frequently occurs in computer learning, where efficient algorithms of combining simple classifiers into a complex one (such as boosting and bagging) have attracted a lot of attention. Using Talagrand's concentration inequalities for empirical processes, we obtain new sharper bounds on the generalization error of combined classifiers that take into account both the empirical distribution of "classification margins'' and an "approximate dimensio…
It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, i.e., the rates faster than $n^{-1/2}$. The works on this subject suggested the following two conjectures: (i) the best achievable fast rate is of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge slower than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only the fast, but also the {\it sup…
Georgia Institute of Technology（佐治亚理工学院）；Pennsylvania State University（宾夕法尼亚州立大学）
Adversarial training is a principled approach for training robust neural networks. Despite of tremendous successes in practice, its theoretical properties still remain largely unexplored. In this paper, we provide new theoretical insights of gradient descent based adversarial training by studying its computational properties, specifically on its inductive bias. We take the binary classification task on linearly separable data as an illustrative example, where the loss asymptotically attains its infimum as the parameter diverges to infinity along certain directions. Specifically, we show that w…
A major challenge in modern machine learning is theoretically understanding the generalization properties of overparameterized models. Many existing tools rely on uniform convergence (UC), a property that, when it holds, guarantees that the test loss will be close to the training loss, uniformly over a class of candidate models. Nagarajan and Kolter (2019) show that in certain simple linear and neural-network settings, any uniform convergence bound will be vacuous, leaving open the question of how to prove generalization in settings where UC fails. Our main contribution is proving novel genera…
Queen Mary University of London（伦敦玛丽女王大学）；Information Technologies Institute/Centre for Research and Technology Hellas (CERTH)（希腊信息技术研究所/希腊研究与技术中心）
In this paper, we propose a maximum margin classifier that deals with uncertainty in data input. More specifically, we reformulate the SVM framework such that each training example can be modeled by a multi-dimensional Gaussian distribution described by its mean vector and its covariance matrix -- the latter modeling the uncertainty. We address the classification problem and define a cost function that is the expected value of the classical SVM cost when data samples are drawn from the multi-dimensional Gaussian distributions that form the set of the training examples. Our formulation approxim…
Hong Kong University of Science and Technology（香港科技大学）
Margin enlargement over training data has been an important strategy since perceptrons in machine learning for the purpose of boosting the robustness of classifiers toward a good generalization ability. Yet Breiman (1999) showed a dilemma that a uniform improvement on margin distribution does NOT necessarily reduces generalization errors. In this paper, we revisit Breiman's dilemma in deep neural networks with recently proposed spectrally normalized margins, from a novel perspective based on phase transitions of normalized margin distributions in training dynamics. Normalized margin distributi…
Consider the binary classification problem of predicting a target variable $Y$ from a discrete feature vector $X = (X_1,...,X_d)$. When the probability distribution $\mathbb{P}(X,Y)$ is known, the optimal classifier, leading to the minimum misclassification rate, is given by the Maximum A-posteriori Probability decision rule. However, estimating the complete joint distribution $\mathbb{P}(X,Y)$ is computationally and statistically impossible for large values of $d$. An alternative approach is to first estimate some low order marginals of $\mathbb{P}(X,Y)$ and then design the classifier based o…
Miami University（迈阿密大学）；University of Alabama（阿拉巴马大学）
Empirical evidence shows that ensembles, such as bagging, boosting, random and rotation forests, generally perform better in terms of their generalization error than individual classifiers. To explain this performance, Schapire et al. (1998) developed an upper bound on the generalization error of an ensemble based on the margins of the training data, from which it was concluded that larger margins should lead to lower generalization error, everything else being equal. Many other researchers have backed this assumption and presented tighter bounds on the generalization error based on either the…
Miami University（迈阿密大学）；University of Alabama（阿拉巴马大学）
Boosting and other ensemble methods combine a large number of weak classifiers through weighted voting to produce stronger predictive models. To explain the successful performance of boosting algorithms, Schapire et al. (1998) showed that AdaBoost is especially effective at increasing the margins of the training data. Schapire et al. (1998) also developed an upper bound on the generalization error of any ensemble based on the margins of the training data, from which it was concluded that larger margins should lead to lower generalization error, everything else being equal (sometimes referred t…
University of Georgia（佐治亚大学）；Zhongnan University of Economics and Law（中南财经政法大学）
Large-margin classifiers are popular methods for classification. We derive the asymptotic expression for the generalization error of a family of large-margin classifiers in the limit of both sample size $n$ and dimension $p$ going to $\infty$ with fixed ratio $α=n/p$. This family covers a broad range of commonly used classifiers including support vector machine, distance weighted discrimination, and penalized logistic regression. Our result can be used to establish the phase transition boundary for the separability of two classes. We assume that the data are generated from a single multivariat…
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We prove bounds on the population risk of the maximum margin algorithm for two-class linear classification. For linearly separable training data, the maximum margin algorithm has been shown in previous work to be equivalent to a limit of training with logistic loss using gradient descent, as the training error is driven to zero. We analyze this algorithm applied to random data including misclassification noise. Our assumptions on the clean data include the case in which the class-conditional distributions are standard normal distributions. The misclassification noise may be chosen by an advers…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Ben-Gurion University（本-古里安大学）
We analyze a family of supervised learning algorithms based on sample compression schemes that are stable, in the sense that removing points from the training set which were not selected for the compression set does not alter the resulting classifier. We use this technique to derive a variety of novel or improved data-dependent generalization bounds for several learning algorithms. In particular, we prove a new margin bound for SVM, removing a log factor. The new bound is provably optimal. This resolves a long-standing open question about the PAC margin bounds achievable by SVM.
Deep Copula Classifier: Theory, Consistency, and Empirical Evaluation
Department of Mathematics, University of Louisiana at Lafayette（路易斯安那大学拉福莱分校数学系）；Department of Statistics, Formerly of Presidency College, Kolkata, India（印度加尔各答 Presidency College 统计学系）
Numbers and numerical vectors account for a large portion of data. However, recently the amount of string data generated has increased dramatically. Consequently, classifying string data is a common problem in many fields. The most widely used approach to this problem is to convert strings into numerical vectors using string kernels and subsequently apply a support vector machine that works in a numerical vector space. However, this non-one-to-one conversion involves a loss of information and makes it impossible to evaluate, using probability theory, the generalization error of a learning mach…
In this work, we study a new approach to optimizing the margin distribution realized by binary classifiers. The classical approach to this problem is simply maximization of the expected margin, while more recent proposals consider simultaneous variance control and proxy objectives based on robust location estimates, in the vein of keeping the margin distribution sharply concentrated in a desirable region. While conceptually appealing, these new approaches are often computationally unwieldy, and theoretical guarantees are limited. Given this context, we propose an algorithm which searches the h…
Convex Relaxation for Solving Large-Margin Classifiers in Hyperbolic Space
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Center for Brain Science, Harvard University（哈佛大学脑科学中心）；Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University（哈佛大学肯普ner自然与人工智能研究所）
We propose Very Simple Classifier (VSC) a novel method designed to incorporate the concepts of subsampling and locality in the definition of features to be used as the input of a perceptron. The rationale is that locality theoretically guarantees a bound on the generalization error. Each feature in VSC is a max-margin classifier built on randomly-selected pairs of samples. The locality in VSC is achieved by multiplying the value of the feature by a confidence measure that can be characterized in terms of the Chebichev inequality. The output of the layer is then fed in a output layer of neurons…
Modern machine learning classifiers often exhibit vanishing classification error on the training set. They achieve this by learning nonlinear representations of the inputs that maps the data into linearly separable classes. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data. We consider a stylized setting in which data $(y_i,{\boldsymbol x}_i)$, $i\le n$ are i.i.d. with ${\boldsymbol x}_i\sim\mathsf{N}({\boldsymbol 0},{\boldsymbol Σ})$ a $p$-dimensional Gaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distribution de…
Harbin Institute of Technology（哈尔滨工业大学）；Sun Yat-sen University（中山大学）
The generalization error bound of support vector machine (SVM) depends on the ratio of radius and margin, while standard SVM only considers the maximization of the margin but ignores the minimization of the radius. Several approaches have been proposed to integrate radius and margin for joint learning of feature transformation and SVM classifier. However, most of them either require the form of the transformation matrix to be diagonal, or are non-convex and computationally expensive. In this paper, we suggest a novel approximation for the radius of minimum enclosing ball (MEB) in feature space…
We develop a general framework for margin-based multicategory classification in metric spaces. The basic work-horse is a margin-regularized version of the nearest-neighbor classifier. We prove generalization bounds that match the state of the art in sample size $n$ and significantly improve the dependence on the number of classes $k$. Our point of departure is a nearly Bayes-optimal finite-sample risk bound independent of $k$. Although $k$-free, this bound is unregularized and non-adaptive, which motivates our main result: Rademacher and scale-sensitive margin bounds with a logarithmic depende…
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Univ. Grenoble Alpes（格勒诺布尔-阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；University of Washington（华盛顿大学）
We propose Rademacher complexity bounds for multiclass classifiers trained with a two-step semi-supervised model. In the first step, the algorithm partitions the partially labeled data and then identifies dense clusters containing $κ$ predominant classes using the labeled training examples such that the proportion of their non-predominant classes is below a fixed threshold. In the second step, a classifier is trained by minimizing a margin empirical loss over the labeled training set and a penalization term measuring the disability of the learner to predict the $κ$ predominant classes of the i…
University of Texas Southwestern Medical Center（德克萨斯大学西南医学中心）
Multiclass problems are often decomposed into multiple binary problems that are solved by individual binary classifiers whose results are integrated into a final answer. Various methods, including all-pairs (APs), one-versus-all (OVA), and error correcting output code (ECOC), have been studied, to decompose multiclass problems into binary problems. However, little study has been made to optimally aggregate binary problems to determine a final answer to the multiclass problem. In this paper we present a convex optimization method for an optimal aggregation of binary classifiers to estimate clas…
We provide a new local class-purity theorem for Lipschitz continuous DNN classifiers. In addition, we discuss how to achieve classification margin for training samples. Finally, we describe how to compute margin p-values for test samples.
Boosting algorithms produce a classifier by iteratively combining base hypotheses. It has been observed experimentally that the generalization error keeps improving even after achieving zero training error. One popular explanation attributes this to improvements in margins. A common goal in a long line of research, is to maximize the smallest margin using as few base hypotheses as possible, culminating with the AdaBoostV algorithm by (R{ä}tsch and Warmuth [JMLR'04]). The AdaBoostV algorithm was later conjectured to yield an optimal trade-off between number of hypotheses trained and the minimal…
Despite the enormous success of machine learning models in various applications, most of these models lack resilience to (even small) perturbations in their input data. Hence, new methods to robustify machine learning models seem very essential. To this end, in this paper we consider the problem of binary classification with adversarial perturbations. Investigating the solution to a min-max optimization (which considers the worst-case loss in the presence of adversarial perturbations) we introduce a generalization to the max-margin classifier which takes into account the power of the adversary…
In this article, we study rates of convergence of the generalization error of multi-class margin classifiers. In particular, we develop an upper bound theory quantifying the generalization error of various large margin classifiers. The theory permits a treatment of general margin losses, convex or nonconvex, in presence or absence of a dominating class. Three main results are established. First, for any fixed margin loss, there may be a trade-off between the ideal and actual generalization performances with respect to the choice of the class of candidate decision functions, which is governed b…
Recent works have shown the effectiveness of randomized smoothing as a scalable technique for building neural network-based classifiers that are provably robust to $\ell_2$-norm adversarial perturbations. In this paper, we employ adversarial training to improve the performance of randomized smoothing. We design an adapted attack for smoothed classifiers, and we show how this attack can be used in an adversarial training setting to boost the provable robustness of smoothed classifiers. We demonstrate through extensive experimentation that our method consistently outperforms all existing provabl…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Toyota Technical Institute of Chicago（芝加哥丰田技术研究所）
We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of $\widetilde{\mathcal{O}}(1/t^2)$. This contrasts with a rate of $\mathcal{O}(1/\log(t))$ for standard gradient descent, and $\mathcal{O}(1/t)$ for normalized gradient descent. This momentum-based method is derived via the convex dual of the maximum-margin problem, and specifically by applying Nesterov acceleration to this dual, which manages to result in a simp…
University of Chicago（芝加哥大学）；Harvard University（哈佛大学）
This paper establishes a precise high-dimensional asymptotic theory for boosting on separable data, taking statistical and computational perspectives. We consider a high-dimensional setting where the number of features (weak learners) $p$ scales with the sample size $n$, in an overparametrized regime. Under a class of statistical models, we provide an exact analysis of the generalization error of boosting when the algorithm interpolates the training data and maximizes the empirical $\ell_1$-margin. Further, we explicitly pin down the relation between the boosting test error and the optimal Bay…
Institute of Information Transmission Problems（信息传输问题研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We consider a problem of risk estimation for large-margin multi-class classifiers. We propose a novel risk bound for the multi-class classification problem. The bound involves the marginal distribution of the classifier and the Rademacher complexity of the hypothesis class. We prove that our bound is tight in the number of classes. Finally, we compare our bound with the related ones and provide a simplified version of the bound for the multi-class classification with kernel based hypotheses.
The Price of Robustness: Stable Classifiers Need Overparameterization
Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Aleph Alpha Research（Aleph Alpha研究公司）；University of Tromso（特罗姆瑟大学）；DLR-German Aerospace Center（德国航空航天中心）
In many real-world applications, data is not collected as one batch, but sequentially over time, and often it is not possible or desirable to wait until the data is completely gathered before analyzing it. Thus, we propose a framework to sequentially update a maximum margin classifier by taking advantage of the Maximum Entropy Discrimination principle. Our maximum margin classifier allows for a kernel representation to represent large numbers of features and can also be regularized with respect to a smooth sub-manifold, allowing it to incorporate unlabeled observations. We compare the performa…
University of Edinburgh（爱丁堡大学）；University of Cambridge（剑桥大学）
We derive a new asymptotic expansion for the global excess risk of a local-$k$-nearest neighbour classifier, where the choice of $k$ may depend upon the test point. This expansion elucidates conditions under which the dominant contribution to the excess risk comes from the decision boundary of the optimal Bayes classifier, but we also show that if these conditions are not satisfied, then the dominant contribution may arise from the tails of the marginal distribution of the features. Moreover, we prove that, provided the $d$-dimensional marginal distribution of the features has a finite $ρ$th m…
Boosting has attracted much research attention in the past decade. The success of boosting algorithms may be interpreted in terms of the margin theory. Recently it has been shown that generalization error of classifiers can be obtained by explicitly taking the margin distribution of the training data into account. Most of the current boosting algorithms in practice usually optimizes a convex loss function and do not make use of the margin distribution. In this work we design a new boosting algorithm, termed margin-distribution boosting (MDBoost), which directly maximizes the average margin and…
ETH Zürich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）
Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lo…
University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）
We give a general recipe for derandomising PAC-Bayesian bounds using margins, with the critical ingredient being that our randomised predictions concentrate around some value. The tools we develop straightforwardly lead to margin bounds for various classifiers, including linear prediction -- a class that includes boosting and the support vector machine -- single-hidden-layer neural networks with an unusual \(\erf\) activation function, and deep ReLU networks. Further, we extend to partially-derandomised predictors where only some of the randomness is removed, letting us extend bounds to cases …
Linear classifiers and leaky ReLU networks trained by gradient flow on the logistic loss have an implicit bias towards solutions which satisfy the Karush--Kuhn--Tucker (KKT) conditions for margin maximization. In this work we establish a number of settings where the satisfaction of these KKT conditions implies benign overfitting in linear classifiers and in two-layer leaky ReLU networks: the estimators interpolate noisy training data and simultaneously generalize well to test data. The settings include variants of the noisy class-conditional Gaussians considered in previous work as well as new…
University of Chicago（芝加哥大学）；University of California, Berkeley（加州大学伯克利分校）
This paper provides elementary analyses of the regret and generalization of minimum-norm interpolating classifiers (MNIC). The MNIC is the function of smallest Reproducing Kernel Hilbert Space norm that perfectly interpolates a label pattern on a finite data set. We derive a mistake bound for MNIC and a regularized variant that holds for all data sets. This bound follows from elementary properties of matrix inverses. Under the assumption that the data is independently and identically distributed, the mistake bound implies that MNIC generalizes at a rate proportional to the norm of the interpol…
NTT CS laboratories（NTT计算机科学实验室）；Unbabel（Unbabel公司）；Instituto de Telecomunicações（电信研究所）
This paper studies Fenchel-Young losses, a generic way to construct convex loss functions from a regularization function. We analyze their properties in depth, showing that they unify many well-known loss functions and allow to create useful new ones easily. Fenchel-Young losses constructed from a generalized entropy, including the Shannon and Tsallis entropies, induce predictive probability distributions. We formulate conditions for a generalized entropy to yield losses with a separation margin, and probability distributions with sparse support. Finally, we derive efficient algorithms, making…
We consider the problem of adaptation to the margin in binary classification. We suggest a penalized empirical risk minimization classifier that adaptively attains, up to a logarithmic factor, fast optimal rates of convergence for the excess risk, that is, rates that can be faster than n^{-1/2}, where n is the sample size. We show that our method also gives adaptive estimators for the problem of edge estimation.
It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than $n^{-1/2}$. The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fas…
